Preheat the oven to 350°F (175°C). Grease a 9-inch round cake pan or a similar baking dish.
In a medium bowl, combine the flour, baking powder, baking soda, cinnamon, nutmeg, and salt. Set aside.
In a large mixing bowl, cream together the softened butter and brown sugar until light and fluffy. Add the eggs one at a time, mixing well after each addition. Stir in the vanilla extract.
Gradually add the dry ingredients to the wet mixture, alternating with the milk. Mix until just combined. Fold in the grated carrots and chopped pecans.
Pour the batter into the prepared cake pan and smooth the top. Bake for 30-35 minutes, or until a toothpick inserted into the center comes out clean. Let it cool in the pan for 10 minutes before transferring to a wire rack to cool completely.
While the pudding is cooling, prepare the brown sugar cream. In a medium saucepan over medium heat, combine the heavy cream, brown sugar, and vanilla extract. Stir until the sugar has dissolved and the mixture is heated through, but do not bring it to a boil. Remove from heat and let it cool slightly.
Serve slices of the sticky pudding warm or at room temperature, drizzled with the brown sugar cream.